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Transient random walk in ℤ 2 with stationary orientations

Françoise Pène (2009)

ESAIM: Probability and Statistics

In this paper, we extend a result of Campanino and Pétritis [Markov Process. Relat. Fields 9 (2003) 391–412]. We study a random walk in ℤ 2 with random orientations. We suppose that the orientation of the kth floor is given by ξ k , where ( ξ k ) k ∈ ℤ is a stationary sequence of random variables. Once the environment fixed, the random walk can go either up or down or can stay in the present floor (but moving with respect to its orientation). This model was introduced by Campanino and Pétritis in [Markov Process....

Two algorithms based on Markov chains and their application to recognition of protein coding genes in prokaryotic genomes

Małgorzata Grabińska, Paweł Błażej, Paweł Mackiewicz (2013)

Applicationes Mathematicae

Methods based on the theory of Markov chains are most commonly used in the recognition of protein coding sequences. However, they require big learning sets to fill up all elements in transition probability matrices describing dependence between nucleotides in the analyzed sequences. Moreover, gene prediction is strongly influenced by the nucleotide bias measured by e.g. G+C content. In this paper we compare two methods: (i) the classical GeneMark algorithm, which uses a three-periodic non-homogeneous...

Une variante de l'inégalité de Cheeger pour les chaînes de Markov finies

Laurent Miclo (2010)

ESAIM: Probability and Statistics

Sur un ensemble fini, on s'intéresse aux minorations linéaires du trou spectral d'un noyau markovien réversible, en terme de la constante isopérimétrique associée. On montre que la constante optimale est l'inverse du cardinal de l'ensemble moins un, mais on verra aussi comment il est possible de l'améliorer dans certaines situations particulières (arbres pointés radiaux à nombre fini de générations). Une application des inégalités précédentes est de retrouver immédiatement le comportement...

Uniform mixing time for random walk on lamplighter graphs

Júlia Komjáthy, Jason Miller, Yuval Peres (2014)

Annales de l'I.H.P. Probabilités et statistiques

Suppose that 𝒢 is a finite, connected graph and X is a lazy random walk on 𝒢 . The lamplighter chain X ⋄ associated with X is the random walk on the wreath product 𝒢 ⋄ = 𝐙 2 ≀ 𝒢 , the graph whose vertices consist of pairs ( f ̲ , x ) where f is a labeling of the vertices of 𝒢 by elements of 𝐙 2 = { 0 , 1 } and x is a vertex in 𝒢 . There is an edge between ( f ̲ , x ) and ( g ̲ , y ) in 𝒢 ⋄ if and only if x is adjacent to y in 𝒢 and f z = g z for all z ≠ x , y . In each step, X ⋄ moves from a configuration ( f ̲ , x ) by updating x to y using the transition rule of X and then sampling both...

Unique Bernoulli g -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a g -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique g -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the g -measure.

Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices

Xiao-Dong Zhang, Rong Luo (2002)

Czechoslovak Mathematical Journal

We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.

Why the Kemeny Time is a constant

Karl Gustafson, Jeffrey J. Hunter (2016)

Special Matrices

We present a new fundamental intuition forwhy the Kemeny feature of a Markov chain is a constant. This new perspective has interesting further implications.

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